   Chapter 11, Problem 5PS

Chapter
Section
Textbook Problem

# Distance(a) Find the shortest distance between the point Q ( 2 , 0 , 0 ) and the line determined by the points P 1 ( 0 , 0 , 1 ) and P 2 ( 0 , 1 , 2 ) .(b) Find the shortest distance between the point Q ( 2 , 0 , 0 ) and the line segment joining the points P 1 ( 0 , 0 , 1 ) and P 2 ( 0 , 1 , 2 ) .

a)

To determine

To calculate: The shortest distance between the point Q(2,0,0) and the line determined by the points P1(0,0,1) and P2(0,1,2).

Explanation

Given:

Points as, Q(2,0,0), P1(0,0,1) and P2(0,1,2).

Formula used:

The distance between a point Q and a line in space is:

D=PQ×uu

where u is a direction vector for the line and P is a point on the line.

Calculation:

A line is determined between the points P1(0,0,1) and P2(0,1,2).

Finding the direction vector, u, for the line formed:

u=P2P1=0,1,20,0,1=0,1,1

Finding the vector formed by the stated point and a point, say P1, on this line formed.

P1Q=QP1=2,0,00,0,1=2,0,1

By applying the formula of distance between a point and a line in space:

D=P1Q×uu

b)

To determine

To calculate: The shortest distance between the point Q(2,0,0) and the line segment joining the points P1(0,0,1) and P2(0,1,2).

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